相关论文: Static solutions to symplectic curvature flow in d…

Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-K\"ahler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study…

J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost K\"ahler manifolds generalising K\"ahler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-K\"ahler…

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a…

Let $M$ be a K\"ahler-Einstein surface with positive scalar curvature. If the initial surface is sufficiently close to a holomorphic curve, we show that the mean curvature flow has a global solution and it converges to a holomorphic curve.

In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the K\"ahler angle $\alpha$ of a symplectic translating soliton with $\max…

We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian…

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…

Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with…

Exploring the general analytical solutions to the Euler equations for ideal fluids holds significant theoretical and practical importance. The steady flows in two-dimensional spaces are considered whether there is an analytical solution in…

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebin's long-time existence result for geodesics on…

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its…

We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in Cartesian coordinate systems. We couple the two fluids through the continuity equation where different…

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2,…

Hamilton flows on K\"ahler manifold for which all trajectories are $H$-planar curves (complex analog of geodesics) are considered. These flows are called $H$-planar. The equation which has to obey the Hamiltonian of $H$-planar Hamilton flow…

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We…

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly…

This paper is concerned with a fourth order nonlinear dispersive partial differential equation for closed curve flow on a K\"ahler manifold. The main results is that the initial value problem has a solution locally in time if the K\"ahler…

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…

In this paper we utilize symmetries in order to exhibit exact solutions to Einstein's equation of a perfect fluid on a static manifold all of whose spatial factor belongs to the conformal class of a Riemannian space of constant curvature.